[Paper Review] Hybrid Batch Bayesian Optimization

Javad Azimi, Ali Jalali+1 more|arXiv (Cornell University)|Feb 25, 2012

Advanced Bandit Algorithms Research12 references44 citations

TL;DR

This paper proposes a hybrid batch Bayesian optimization algorithm that dynamically switches between sequential and batch evaluation to balance optimization performance and time efficiency. By using a theoretical stopping criterion based on prediction error, the method achieves up to 78% speedup over sequential Bayesian optimization with minimal performance loss, outperforming existing batch methods like Constant Liar with $ ilde{y} = \hat{\mu}$.

ABSTRACT

Bayesian Optimization aims at optimizing an unknown non-convex/concave function that is costly to evaluate. We are interested in application scenarios where concurrent function evaluations are possible. Under such a setting, BO could choose to either sequentially evaluate the function, one input at a time and wait for the output of the function before making the next selection, or evaluate the function at a batch of multiple inputs at once. These two different settings are commonly referred to as the sequential and batch settings of Bayesian Optimization. In general, the sequential setting leads to better optimization performance as each function evaluation is selected with more information, whereas the batch setting has an advantage in terms of the total experimental time (the number of iterations). In this work, our goal is to combine the strength of both settings. Specifically, we systematically analyze Bayesian optimization using Gaussian process as the posterior estimator and provide a hybrid algorithm that, based on the current state, dynamically switches between a sequential policy and a batch policy with variable batch sizes. We provide theoretical justification for our algorithm and present experimental results on eight benchmark BO problems. The results show that our method achieves substantial speedup (up to %78) compared to a pure sequential policy, without suffering any significant performance loss.

Motivation & Objective

• To address the trade-off between optimization performance and time efficiency in Bayesian optimization when function evaluations are costly.
• To develop a hybrid algorithm that intelligently transitions from sequential to batch mode based on model uncertainty and estimation error.
• To provide a theoretical foundation for analyzing the bias introduced by simulated outcomes in batch Bayesian optimization.
• To improve upon existing batch BO methods like Constant Liar by introducing a principled stopping criterion for batch size selection.
• To empirically validate that the hybrid approach maintains performance close to sequential BO while significantly reducing the number of iterations.

Proposed method

• The algorithm uses a Gaussian Process (GP) to model the unknown function and applies Expected Improvement (EI) as the acquisition function.
• At each iteration, the algorithm selects the next best single point using EI, then estimates its outcome using the GP posterior mean.
• It dynamically decides whether to add another point to the current batch by evaluating a theoretical stopping criterion based on the prediction error of the estimated outcome.
• The stopping criterion is defined as $\gamma_z \theta_x \leq \epsilon$, where $\gamma_z$ measures the sensitivity of the next best point to the estimated outcome, and $\theta_x$ is the estimation error.
• If the condition is met, the algorithm continues adding points to the batch; otherwise, it switches back to sequential mode.
• The method naturally evolves from sequential behavior in early stages (high uncertainty) to batch behavior in later stages (low uncertainty), enabling variable batch sizes.

Experimental results

Research questions

• RQ1 How can we theoretically bound the bias introduced by estimating function outcomes in simulation-based batch Bayesian optimization?
• RQ2 What conditions ensure that adding another point to a batch does not significantly degrade optimization performance compared to sequential selection?
• RQ3 Can a hybrid approach that dynamically adjusts batch size based on model uncertainty outperform fixed-batch and purely sequential Bayesian optimization?
• RQ4 How does the choice of the simulated outcome (e.g., $\hat{y} = \hat{\mu}$) affect the performance and stability of batch selection in Bayesian optimization?
• RQ5 To what extent can a hybrid algorithm achieve both high time efficiency and high optimization accuracy in real-world and synthetic problems?

Key findings

• The proposed hybrid algorithm achieves up to 78% speedup over sequential Bayesian optimization on benchmark problems, significantly reducing the number of iterations required.
• The algorithm maintains optimization performance nearly identical to sequential EI, with no significant performance degradation even at small batch sizes.
• The theoretical analysis shows that the distance between the second experiment in a batch and the true sequential choice is upper-bounded by a term proportional to the square root of the estimation error.
• Setting the simulated outcome to the posterior mean $\hat{\mu}$ in the Constant Liar heuristic leads to a more favorable stopping condition ($\gamma_z \theta_x \leq \epsilon$), justifying its use in practice.
• The $\mu$-Constant Batch variant, which uses $\hat{y} = \hat{\mu}$, performs competitively with the state-of-the-art Matching method but with significantly lower computational complexity.
• Empirical results confirm that the hybrid algorithm effectively transitions from sequential to batch behavior, with early iterations favoring sequential selection due to high uncertainty.